How Thermal Quantum Pressure of free, massive, and bosonic Quasiparticles probes Riemann's Zeta Function

A known integral representation of the quantum pressure of free, massive, and bosonic quasiparticles is considered. Depending on how this integral is evaluated, the pressure can be written, modulo a power times logarithmic correction, as a convergent power series, valid for small masses, or as rapidly convergent infinite sum over modified Bessel functions when mass is large. The derivative of this integral with respect to mass squared can be represented as an optimally truncated, exponentially asymptotic series. This series is employed to analyze the evolution of quasiparticle mass [Formula: see text] in the deconfining phase of SU(2) Yang-Mills thermodynamics down to the critical temperature. Expanding the integral for the quantum pressure in powers of the logarithm of [Formula: see text] yields large-[Formula: see text] coefficients [Formula: see text] which probe Riemann’s zeta function [Formula: see text] within the critical strip 0 [Formula: see text] 1 at large imaginary heights. Appealing to the meromorphic nature of the integrands, we employ a saddle point approximation to estimate the large-n asymptotics of cn by integration along contours of steepest descent. As a result, comparisons between saddle-point estimates and localized integrations along vertical lines are facilitated to test assumed properties of [Formula: see text] within the critical strip.